Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Adjacency-Operator Formulation) (2512.12226v1)

Abstract: Classical spectral graph theory and graph signal processing rely on a symmetry principle: undirected graphs induce symmetric (self-adjoint) adjacency/Laplacian operators, yielding orthogonal eigenbases and energy-preserving Fourier expansions. Real-world networks are typically directed and hence asymmetric, producing non-self-adjoint and frequently non-normal operators for which orthogonality fails and spectral coordinates can be ill-conditioned. In this paper we develop an original harmonic-analysis framework for directed networks centered on the \emph{adjacency} operator. We propose a \emph{Biorthogonal Graph Fourier Transform} (BGFT) built from left/right eigenvectors, formulate directed ``frequency'' and filtering in the non-Hermitian setting, and quantify how asymmetry and non-normality affect stability via condition numbers and a departure-from-normality functional. We prove exact synthesis/analysis identities under diagonalizability, establish sampling-and-reconstruction guarantees for BGFT-bandlimited signals, and derive perturbation/stability bounds that explain why naive orthogonal-GFT assumptions break down on non-normal directed graphs. A simulation protocol compares undirected versus directed cycles (asymmetry without non-normality) and a perturbed directed cycle (genuine non-normality), demonstrating that BGFT yields coherent reconstruction and filtering across asymmetric regimes.

• The paper presents a BGFT framework that rigorously extends harmonic analysis to directed networks using biorthogonal left/right eigenbases.
• It establishes exact analysis/synthesis identities and quantifies stability via explicit sensitivity bounds and perturbation theory.
• The approach enables multiresolution analysis and robust spectral filtering in non-normal regimes, guiding filter design for complex networks.

Harmonic Analysis on Directed Graphs via Biorthogonal Bases: An Adjacency-Operator Framework

Introduction and Motivation

The paper addresses a critical foundational gap in spectral graph theory and graph signal processing by rigorously extending harmonic analysis to directed (asymmetric) networks. Classical methods are intrinsically reliant on operator symmetry—orthogonal eigenbases, energy-preserving spectral decompositions, and robust filtering are predicated on underlying self-adjointness, as is guaranteed in undirected graphs. However, true directed networks yield asymmetric, often non-normal adjacency operators, whose spectral properties diverge sharply from the undirected case. This asymmetry and potential non-normality destroy orthogonality, causing instability and ill-conditioning in spectral domains, thus invalidating naive extensions of the classical Graph Fourier Transform (GFT).

The work develops a complete theoretical framework for adjacency-based harmonic analysis on directed networks, leveraging the machinery of biorthogonal (left/right) spectral bases. This biorthogonal Graph Fourier Transform (BGFT) provides exact analysis/synthesis identities, spectral filtering, and multiresolution decompositions for directed graphs, with quantitative stability guarantees even in non-normal regimes.

Biorthogonal Graph Fourier Transform (BGFT) Formulation

Construction and Exactness

Given a diagonalizable adjacency matrix $A$, left and right eigenvectors are constructed so that $A v_k = \lambda_k v_k$ and $u_k^* A = \lambda_k u_k^*$, with $u_k^* v_\ell = \delta_{k\ell}$. The adjoint system $U^* = V^{-1}$ underpins BGFT analysis ($\widehat{x} = U^* x$) and synthesis ($x = V \widehat{x}$). This construction ensures algebraic exactness—signal representations are perfectly invertible, and spectral filtering is precisely characterized in the BGFT domain.

The BGFT generalizes directly to non-diagonalizable adjacency operators via Jordan chains: the decomposition $A = V J V^{-1}$ (with $J$ block diagonal in Jordan blocks) supports generalized biorthogonal analysis/synthesis. Filtering in this setting, significant for real-world network data, is handled via structured polynomial action on Jordan blocks.

Quantification of Asymmetry and Non-Normality

The paper distinguishes structural asymmetry (e.g., directionality) from non-normality (quantified by functional commutator norms). The Frobenius-based asymmetry index vanishes if and only if $A$ is symmetric, while the departure-from-normality functional gauges how far $A$ is from satisfying $A A^* = A^* A$. This allows one to probe when asymmetry induces numerical instability—a key theoretical and practical advance not addressed in prior literature.

The authors emphasize, and empirically demonstrate, that directed graphs can be highly asymmetric yet perfectly normal (e.g., the directed cycle), in which case the BGFT inherits well-conditioned bases ($\kappa(V) = 1$), unitary spectral geometry, and stable reconstruction. Genuine non-normality alone destroys these assurances and underlies spectral instability.

Energy Geometry and Generalized Parseval Identity

Unlike the symmetric case (Parseval’s identity), where energy is preserved in the coefficient domain, BGFT works with non-unitary bases. The correct energy metric in the spectral domain is the Gram matrix $G = V^* V$, yielding the identity $\|x\|_2^2 = \|\widehat{x}\|_G^2$. Filtering operations and variational estimates must therefore be formulated in the geometry induced by $G$.

Stability, Conditioning, and Sampling

Perturbation Theory

The stability of spectral coordinates under directed operators is governed by the eigenbasis condition number $\kappa(V) = \|V\|_2 \|V^{-1}\|_2$. The paper derives explicit Bauer-Fike-type bounds for eigenvalue movement under perturbation, establishing the tight connection between non-normality and instability: eigenvalues and reconstructed signals are robust only when $\kappa(V)$ is not excessively large.

Sampling and Reconstruction

For BGFT-bandlimited signals ($x \in \operatorname{span}(V_\Omega)$), exact recovery from sampled nodes is possible via a (possibly ill-conditioned) linear inversion. The paper proves explicit sensitivity bounds: noise amplification is proportional to $\|V_\Omega\|_2/\sigma_{\min}(P_M V_\Omega)$, making the conditioning of both eigenbases and the sampling operator central to any practical implementation on directed graphs.

Empirical Protocol and Asymmetry-Non-Normality Archetypes

A rigorous computational protocol contrasts three regimes:

• Undirected cycles: symmetric and normal ($\kappa(V) = 1$, stable).
• Directed cycles: asymmetric but normal, with unitary and stable BGFT (contradicting the naive “asymmetry = ill-conditioning” heuristic).
• Perturbed directed cycles: asymmetric and non-normal, where $\kappa(V)$ grows and instability appears.

These experiments operationalize the theoretical insights, showing why filter and transform design for directed networks cannot rely solely on symmetry-inspired techniques.

Multiresolution and Wavelet Framework

The paper outlines the extension of BGFT to wavelet/multiresolution analysis on directed networks. In contrast to orthogonal filter banks, biorthogonal perfect reconstruction conditions ($H_0\widetilde{H}_0 + H_1\widetilde{H}_1 = I$) must be satisfied in the spectral domain, reflecting the dual basis structure inherent to BGFT. Adapting hierarchical analysis to this setting lays the groundwork for future developments in localized, multiscale processing on directed graphs.

Implications and Open Problems

This work reframes spectral graph analysis for directed networks by foregrounding biorthogonality and stability under non-normality. The theoretical perspective makes precise the conditions under which harmonic analysis, filtering, and sampling are robust, and the empirical results demonstrate the mathematical tightness of these conditions.

Practical implications span all domains where directed graphs arise: citation networks, social dynamics, transportation networks, and biological signaling pathways, which are fundamentally non-symmetric. The techniques and stability estimates guide the design of spectral transforms and filter banks appropriate for large-scale, potentially non-normal directed network data.

Three concrete avenues for future research are articulated:

1. Selection of shift operators optimizing BGFT stability for a given topology,
2. Scalable computation of BGFT representations (for large, sparse graphs) that avoid explicit Jordan decompositions,
3. Formulation of an uncertainty principle in the non-Hermitian (biorthogonal) setting, relating spectral and node-domain localization via conditioning.

Conclusion

The paper introduces a mathematically rigorous, practically motivated BGFT framework enabling the extension of harmonic analysis to directed networks, handling biorthogonality, non-normality, and their stability implications in full detail. The theoretical contributions, supported by explicit sensitivity bounds and a reproducible computational protocol, provide a robust foundation for future analysis, filter bank design, and efficient signal processing on complex directed graphs. Further development of multiscale and scalable BGFT methods promises substantial impact on the computational analysis of directed network data in both theoretical and applied settings.